Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing

Reservoir computers are powerful tools for chaotic time series prediction. They can be trained to approximate phase space flows and can thus both predict future values to a high accuracy and reconstruct the general properties of a chaotic attractor without requiring a model. In this work, we show that the ability to learn the dynamics of a complex system can be extended to systems with multiple co-existing attractors, here a four-dimensional extension of the well-known Lorenz chaotic system. We demonstrate that a reservoir computer can infer entirely unexplored parts of the phase space; a properly trained reservoir computer can predict the existence of attractors that were never approached during training and, therefore, are labeled as unseen. We provide examples where attractor inference is achieved after training solely on a single noisy trajectory.
Reservoir computing is a brain-inspired machine learning scheme that can be used to mimic dynamical systems. Reservoir computers can be trained to learn the characteristics of a target dynamical system purely from a sample time series. In particular, properly trained autonomous reservoir computers can act as surrogate systems while still preserving many properties of the original ground truth such as the largest Lyapunov exponents, embedded unstable periodic orbits, or correlation measures. Importantly, dynamical systems can exhibit more than just a single stable long-term behavior, called an attractor. A common scenario is the existence of pairs of symmetric solutions, but more complex co-existences can also often be found. Systems with multiple attractors are called multistable. Here, we provide examples where a reservoir computer is able to learn the various attractors of a multistable system. We feed the reservoir just a single noisy trajectory of one of the attractors, while the other attractors remain outside of the training data range. Then, in separate autonomous operation, the trained reservoir is able to reproduce and, therefore, infer the existence and shape of these unseen attractors.